Math 312 Linear Algebra
Summer 2016, Session II
Instructor
Email
Course website
Lectures
Office hours
Extra help
Sebastian Moore
moose@math.upenn.edu
www.math.upenn.edu/~moose/teaching.html and Canvas
MTWR 10AM12:10PM in DRL 4C6
MW 1-2 or by appointment i
Springer-Verlag Berlin Heidelberg GmbH
Jean Jacod
Philip Protter
Probability
Essentials
Second Edition
123
Philip Protter
Jean Jacod
Universite de Paris
Laboratoire de Probabilites
4, place Jussieu - Tour 56
75252 Paris Cedex 05, France
School of Operatio
s
i
1
i
31
i
J
J
J
NYU
MATH-Uh. 110
Linear algebra:
Midterm Exam 1
Spring 2016
Name: E UVK CD & bELH/OTYG.
This exam is scheduled for 100 minutes. No calcuiators, notes, or other outside mate—
riaEs are permitted .The exam is worth 70 points
Probl
l
t
QUIZ #2
Write your name at the top of the page. Put the answer in the provided box, or
circle the answer. Do not show any working.
(1) (1pt) Compute the matrix product
120
021
tor—lot
n—w—tr—l
(b) [21 3) (d) _1 3]
(2) (lpt) Let A be a matrix with (1
QUIZ #1
Write your name at the top of the page. Put the answer in the provided box, or ‘
circle the answer. Do not Show any working.
(1) (2 points) In this graphic, u and v axe unit vectors.
y
E
E
E
E
E
Order the quantities below from least to greates
QUIZ #4
Write your name at the top of the page. Put the answer in the provided box, or
circle the answer. Do not Show any working.
(1) (No partial credit given.)
Suppose A is s 51 X 37 matrix with rank 25.
n / N '
e 1 point The left null space of A is a E
Solutions to Exercises
51
36 Q; R!
! D
" q r.A/ produces from A (m by n of rank n) a full-size square Q D Q1 Q2 !
R
. The columns of Q1 are the orthonormal basis from Gram-Schmidt of the
0
column space of A. The m ! n columns of Q2 are an orthonormal basi
Solutions to Exercises
42
23 As in Problem 22: Row space basis .3; 0; 3/; .1; 1; 2/; column space basis .1; 4; 2/,
24
25
26
27
28
29
30
31
32
.2; 5; 7/; the rank of (3 by 2) times (2 by 3) cannot be larger than the rank of either
factor, so rank ! 2 and t
Solutions to Exercises
46
25 The column space of P will be S . Then r D dimension of S D n.
26 A!1 exists since the rank is r D m. Multiply A2 D A by A!1 to get A D I .
27 If AT Ax D 0 then Ax is in the nullspace of AT . But Ax is always in the column spa
Solutions to Exercises
21
46 Inverting the identity A.I C BA/ D .I C AB/A gives .I C BA/!1 A!1 D A!1 .I C
AB/!1 . So I CBA and I CAB are both invertible or both singular when A is invertible.
(This remains true also when A is singular : Problem 6.6.19 wil
Solutions to Exercises
z2 ! z1 D b1
10 z3 ! z2 D b2
0 ! z3 D b3
7
z1 D !b1 ! b2 ! b3
z2 D
!b2 ! b3
z3 D
!b3
"
# " #
!1 !1 !1
b1
0 !1 !1
b2 D !1 b
D
0
0 !1
b3
11 The forward differences of the squares are .t C 1/2 ! t 2 D t 2 C 2t C 1 ! t 2 D 2t C 1.
Diffe
Solutions to Exercises
2
Problem Set 1.1, page 8
1 The combinations give (a) a line in R3
2
3
4
5
6
7
8
9
10
11
(b) a plane in R3 (c) all of R3 .
v C w D .2; 3/ and v ! w D .6; !1/ will be the diagonals of the parallelogram with v
and w as two sides going
Solutions to Exercises
11
23 If ordinary elimination leads to x C y D 1 and 2y D 3, the original second equation
24
25
26
27
28
29
30
31
32
could be 2y C `.x C y/ D 3 C ` for any `. Then ` will be the multiplier to reach
2y D 3.
!
"
a 2
Elimination fails
Scientic Computing:
Dense Linear Systems
Aleksandar Donev
Courant Institute, NYU1
donev@courant.nyu.edu
1 Course
MATH-GA.2043 or CSCI-GA.2112, Spring 2012
February 9th, 2012
A. Donev (Courant Institute)
Lecture III
2/9/2011
1 / 34
Outline
1
Linear Algebra
Scientic Computing
Sources of Error
Aleksandar Donev
Courant Institute, NYU1
donev@courant.nyu.edu
1 Course
MATH-GA.2043 or CSCI-GA.2112, Spring 2012
February 2nd, 2011
A. Donev (Courant Institute)
Lecture II
2/2/2011
1 / 46
Outline
1
Sources of Error
2
P
Scientic Computing
Numerical Computing
Aleksandar Donev
Courant Institute, NYU1
donev@courant.nyu.edu
1 Course
MATH-GA.2043 or CSCI-GA.2112, Spring 2012
January 26th, 2012
A. Donev (Courant Institute)
Lecture I
1/26/2012
1 / 34
Outline
1
Logistics
2
Condi
Scientic Computing, Spring 2012
Assignment I: Numerical Computing
Aleksandar Donev
Courant Institute, NYU, donev@courant.nyu.edu
Feb 1st, 2012
Due by Thursday Feb. 16th, 2012
1
[10 points] Floating-Point Exceptions
Using MATLAB, do some simple calculation
Scientic Computing, Spring 2012
Assignment III: SVD, Nonlinear Equations and Optimization
Aleksandar Donev
Courant Institute, NYU, donev@courant.nyu.edu
Posted March 1st, 2012
Due by Sunday March 25th, 2012
A total of 100 points is possible (100 points =
Scientic Computing, Spring 2012
Assignment IV: Function Approximation
Aleksandar Donev
Courant Institute, NYU, donev@courant.nyu.edu
Posted March 28th, 2012
Due Thursday April 12th, 2012
A total of 100 points is possible (100 points = A+) with up to 25 ex
Scientic Computing, Spring 2012
Assignment V: Monte Carlo
Aleksandar Donev
Courant Institute, NYU, donev@courant.nyu.edu
Posted April 11th, 2012
Due: Sunday April 22nd, 2012
A total of 75 points is possible.
1
[75pts] Monte Carlo in One Dimension
We consi
Scientic Computing, Spring 2012
Assignment II: Linear Systems
Aleksandar Donev
Courant Institute, NYU, donev@courant.nyu.edu
February 16th, 2012
Due by Thursday March 1st
A total of 100 points is possible. Make sure to follow good programming practices in
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 1 (due Sep. 25, 2014)
Please attempt all homework problems and hand them in by the due date (or earlier!),
A
A
preferably using LTEX typesetting (try LyX if you are not yet familiar with L
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 5 (due Dec. 4, 2014)
1. [1+2pt] We dene an inner product in RN given by
(v, w)K := v T Kw,
where K is the matrix dened in problem 7b on Assignment 3.
(a) Show that (, )K denes an inner prod
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 6 (due Dec. 18, 2014)
0. Problems 6 and 7 from Homework 5.
1. [2+2+2pt] We have seen that the CG method for solving Ax = b, (A Rnn symmetric
and positive) is optimal in the following sense:
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 4 (due Nov. 6, 2014)
1. [2pt] For n, m 1 and a eld K, let A K nm and B K mn . Show that the nonzero
eigenvalues of AB and BA are the same and express the eigenvectors of AB in terms of
thos
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 3 (due Oct. 23, 2014)
1. [3pt] Let
B=
2
.
2 3+
For which values of R is there a symmetric matrix A such that
AB + BA =
4 7
?
7 12
2. [3pt] Which of the following statements are true for any
Fall 2014: MATH-GA: 2111.001
Linear Algebra (one term)
Assignment 2 (due Oct. 9, 2014)
1. Find the change-of-basis matrix P from the canonical basis S = cfw_e1 , e2 in R2 to the basis
S = cfw_v 1 , v 2 , with
2
4
v1 =
, v2 =
.
3
5
1
with respect to the b
Linear Algebra (one term)
MATH-GA 2111.001
Georg Stadler
Courant Institute, NYU
stadler@cims.nyu.edu
Fall 2014, Thursdays, 9:0010:50AM
September 4, 2014
1 / 13
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